Question

Two beams of light having intensities $I$ and $4I$ interfere to produce a fringe pattern on a screen. The phase difference between the beams is $\pi / 2$ at point A and $\pi$ at point B. Then the difference between the resultant intensities at A and B is

• A. $4I$
• B. $5I$
• C. $2I$
• D. $3I$

Hint:

When phase difference is $\pi/2$, the waves are orthogonal and the interference term disappears, leaving just the simple sum of individual intensities ($I + 4I = 5I$). When phase difference is $\pi$, the waves interfere destructively to form a minimum, which can be found quickly using $(\sqrt{I_2} - \sqrt{I_1})^2 = (2\sqrt{I} - \sqrt{I})^2 = I$. The difference is simply $5I - I = 4I$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Two coherent light waves with individual intensities $I_1 = I$ and $I_2 = 4I$ overlap to form an interference pattern.
At a specific point A on the screen, their phase difference is $\phi_A = \pi / 2$. At another point B, their phase difference is $\phi_B = \pi$. We need to compute the absolute difference between the total resultant intensities at these two locations ($I_A - I_B$).

Step 2: Key Formula or Approach:
The general mathematical formula for the resultant intensity $I_{res}$ produced by the interference of two wave fields is given by:
$$I_{res} = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$ where $\phi$ represents the phase difference at that spatial position.

Step 3: Detailed Explanation:
Let's first calculate the resultant intensity at point A where $\phi_A = \pi / 2$:
$$I_A = I + 4I + 2\sqrt{I \cdot 4I}\cos\left(\frac{\pi}{2}\right)$$ Since $\cos(\pi / 2) = 0$, the entire interference term drops to zero:
$$I_A = I + 4I + 0 = 5I$$ Next, calculate the resultant intensity at point B where $\phi_B = \pi$:
$$I_B = I + 4I + 2\sqrt{I \cdot 4I}\cos(\pi)$$ Since $\cos(\pi) = -1$ and $2\sqrt{4I^2} = 2(2I) = 4I$:
$$I_B = 5I + 4I(-1) = 5I - 4I = I$$ Now, calculate the requested difference between the resultant intensities at points A and B:
$$\Delta I = I_A - I_B = 5I - I = 4I$$

Step 4: Final Answer:
The difference between the resultant intensities is $4I$, which corresponds to option (A).